To clean dirty greasy hands, slobber lots of skin lotion onto them, work it around, and then wipe it off onto a rag. Most of the dirty grease will come off with the lotion.
Technique demonstrated by Star.
To make it work even better, before starting your dirty work, rub lots of lotion into your hands .
Especially work it under your nails. In the cleanup phase, an old toothbrush works great for cleaning fingernails.